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Formula Sheet
• Conservation of probability
∂
∂
ρ(x, t) +
J(x, t) = 0
∂t
∂x
~
∂ ∗
2
∗ ∂
ρ(x, t) = |ψ(x, t)| ; J(x, t) =
ψ
ψ−ψ ψ
2im
∂x
∂x
• Variational principle:
dx ψ ∗ (x)Hψ(x)
Egs ≤ R
, for all ψ(x)
dxψ ∗ (x)ψ(x)
R
• Spin-1/2 particle:
Stern-Gerlach :
µB =
~,
H = −~µ · B
e~
,
2me
~µ = g
~µe = −2 µB
e~ 1 ~
~
S = γS
2m ~
~
S
,
~
1
0
In the basis |1i ≡ |z; +i = |+i =
, |2i ≡ |z; −i = |−i =
0
1
~
Si = σi
2
σx =
0 1
1 0
[σi , σj ] = 2iǫijk σk →
σi σj = δij I + iǫijk σk
; σy =
0 −i
i 0
; σz =
Si , Sj = i~ ǫijk Sk
1 0
0 −1
(ǫ123 = +1)
→ (~σ · ~a)(~σ · ~b) = ~a · ~b I + i~σ · (~a × ~b)
eiMθ = 1 cos θ + iM sin θ , if M2 = 1
~a exp i~a · ~σ = 1 cos a + i~σ ·
sin a ,
a
a = |a|
~
exp(iθσ3 ) σ1 exp(−iθσ3 ) = σ1 cos(2θ) − σ2 sin(2θ)
exp(iθσ3 ) σ2 exp(−iθσ3 ) = σ2 cos(2θ) + σ1 sin(2θ) .
~ = nx Sx + ny Sy + nz Sz = ~ ~n · ~σ .
S~n = ~n · S
2
(nx , ny , nz ) = (sin θ cos φ, sin θ sin φ, cos θ) ,
|~n; +i =
cos(θ/2)|+i +
S~n |~n; ±i = ±
sin(θ/2) exp(iφ)|−i
|~n; −i = − sin(θ/2) exp(−iφ)|+i + cos(θ/2)|−i
1
~
|n;
~ ±i
2
• Bras and kets: For an operator Ω and a vector v, we write |Ωvi ≡ Ω|vi
Adjoint: hu|Ω† vi = hΩu|vi
|α1 v1 + α2 v2 i = α1 |v1 i + α2 |v2 i ←→ hα1 v1 + α2 v2 | = α1∗ hv1 | + α2∗ hv2 |
• Complete orthonormal basis |ii
hi|ji = δij ,
1=
X
|iihi|
X
Ωij |iihj |
i
Ωij = hi|Ω|ji ↔ Ω =
i,j
†
hi|Ω |ji = hj|Ω|ii∗
Ω hermitian: Ω† = Ω,
• Matrix M is normal ([M, M † ] = 0) ←→
U unitary: U † = U −1
unitarily diagonalizable.
˜
• Position and momentum representations: ψ(x) = hx|ψi ; ψ(p)
= hp|ψi ;
Z
x̂|xi = x|xi , hx|yi = δ(x − y) , 1 = dx |xihx| , x̂† = x̂
Z
p̂|pi = p|pi , hq|pi = δ(q − p) , 1 = dp |pihp| , p̂† = p̂
Z
Z
ipx 1
ipx 1
dx exp −
ψ (x)
exp
;
ψ̃(p) = dxhp|xihx|ψi = √
hx|pi = √
~
~
2π~
2π~
~ d n
d n
~
n
n
hx|p̂ |ψi =
ψ(x) ;
hp|x̂ |ψi = i~
ψ̃(p) ;
[p̂, f (x̂)] = f ′ (x̂)
i dx
dp
i
Z ∞
1
exp(ik x)dx = δ(k)
2π −∞
• Generalized uncertainty principle
(∆A)2 ≡ h(A − hAi)2 i = hA2 i − hAi2
2
1
(∆A) (∆B) ≥ hΨ| [A, B]|Ψi
2i
2
2
∆x ∆p ≥
~
2
1 x2 ∆
~
∆x = √ and ∆p = √
for a gaussian wavefuntion ψ ∼ exp −
2 ∆2
2
2∆
r
Z +∞
π
dx exp −ax2 =
a
−∞
Time independent operator Q :
∆H∆t ≥
~
,
2
2
d
i
hQi = h [H, Q]i
dt
~
∆Q
∆t ≡ dhQi dt
• Commutator identities
[A, BC] = [A, B]C + B[A, C] ,
1
1
eA Be−A = B + [A, B] + [A, [A, B]] + [A, [A, [A, B]]] + . . . ,
2
3!
eA Be−A = B + [A, B] ,
if [[A, B], A] = 0 ,
[ B , eA ] = [ B , A ]eA ,
if [[A, B], A] = 0
1
1
eA+B = eA eB e− 2 [A,B] = eB eA e 2 [A,B] ,
if [A, B] commutes with A and with B
• Harmonic Oscillator
1 2 1
1
2 2
Ĥ =
p̂ + mω x̂ = ~ω N̂ +
, N̂ = ↠â
2m
2
2
r
r
mω
ip̂
mω
ip̂
†
â =
x̂ +
, â =
x̂ −
,
2~
mω
2~
mω
r
r
~
mω~ †
†
(â + â ) , p̂ = i
(â − â) ,
x̂ =
2mω
2
[x̂, p̂] = i~ , [â, ↠] = 1 ,
[N̂ , â ] = −â ,
ˆ , ↠] = ↠.
[N
1
|ni = √ (a† )n |0i
n!
Ĥ|ni = En |ni = ~ω n +
1
|ni , N̂ |ni = n|ni , hm|ni = δmn
2
√
√
n + 1|n + 1i , â|ni = n|n − 1i .
mω 1/4
mω ψ0 (x) = hx|0i =
exp −
x2 .
π~
2~
↠|ni =
3
MIT OpenCourseWare
http://ocw.mit.edu
8.05 Quantum Physics II
Fall 2013
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